Hey there
Several ways to do that, below you can find my preferred approach, some artists might do it slightly different though:

In general, when you create a project in UE or Unity you set the location where all the project files will be saved.
Within that folder, there will be a folder called "Assets" (Unity5) or "Content" (UE). Within those folders I usually make new folders for my own assets.

My preference is to just use Windows Explorer and copy/move my 3D files (fbx file, textures) in there / directly export my fbx/textures to that project folder, when you get back into the engine it will notice those added files and import them to your project. Then you can just drag them into your scene, and you're good to go^^

Here's a nice video for UE4, but there are also videos for Unity around the web Otherwise, check out official documentation pages^^

In the tutorial mentionned by @Laggtastic the tutor is exporting one solid object. What about a mesh that consists of multiple pieces that has been animated and you need to have it as one mesh in the future. If we merge all pieces into one and export to the engine - will our annimation work ?.